Work

On entropy of pure mixing maps on dendrites

Year: 2026

Type: article

Abstract: For every dollar sign 0 less than alpha less than or equals infinity dollar sign $0 \lt \alpha\le\infty$ 0 < α ≤ ∞ $ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy dollar sign alpha dollar sign $\alpha$ $ α $ . It has been previously shown by Špitalský that there are exact maps on the Gehman dendrite with arbitrarily low positive topological entropy. Together, these results show that the entropy of maps on the Gehman dendrite does not exhibit the paradoxical behaviour reported for graph maps, where the infimum of the topological entropy of exact maps is strictly smaller than the infimum of the entropy of pure mixing maps. The latter result, stated in terms of popular notions of chaos, says that for maps on graphs, lower entropy implies stronger Devaney chaos. The conclusion of this paper says that lower entropy does not force stronger chaos for maps of the Gehman dendrite.

Source: Proceedings of the Royal Society of Edinburgh Section A Mathematics

DOI: 10.1017/prm.2026.10155

Authors: Dominik Kwietniak, Piotr Oprocha, Jakub Tomaszewski

Institutions: Jagiellonian University, University of Ostrava, University of Maryland, College Park, AGH University of Krakow

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funder: European Commission